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Mathematics > Geometric Topology

Title:
Properties of knots preserved by cabling

Abstract: We examine geometric properties of a knot J that are unchanged by taking a
(p,q)-cable K of J. Specifically, we relate w(K) to w(J), where w(K) is the
width of K in the sense of Gabai. We use this information to demonstrate that
thin position is a minimal bridge position of J if and only if the same is true
for K, and more generally we show that any thin position of K is an "obvious"
cabling of a thin position of J. We conclude by proving that J is meridionally
small (mp-small) if and only if K is meridionally small (mp-small), and if J is
mp-small and every non-minimal bridge position of J is stabilized, then the
same is true for K.